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Warm dark matter has been strongly constrained in recent years as the sole component of dark matter. However, a less-explored alternative is that dark matter consists of a mixture of warm and cold dark matter (MWDM). In this work, we use observations of Milky Way satellite galaxies to constrain MWDM scenarios where the formation of small-scale structure is suppressed either by generic thermal relic warm dark matter or a sterile neutrino produced through the Shi-Fuller mechanism. To achieve this, we model satellite galaxies by combining numerical simulations with semianalytical models for the subhalo population, and use a galaxy-halo connection model to match galaxies onto dark matter subhalos. By comparing the number of satellites predicted by MWDM models to the observed satellite population from the Dark Energy Survey and Pan-STARRS1, we constrain the fraction of warm dark matter, 𝑓WDM, as a function of its mass, 𝑚WDM. We exclude dark matter being composed entirely of thermal relic warm dark matter with 𝑚WDM ≤6.6  keV at a posterior ratio of 10∶1, consistent with previous works. However, we find that warm dark matter with smaller mass is allowed when mixed with cold dark matter, and that the 𝑓WDM constraints strengthen with decreasing 𝑚WDM until they plateau at 𝑓WDM ≲0.45 for 𝑚WDM ≲1.5  keV. Likewise, in the case of a sterile neutrino with mass of 7 keV produced through the Shi-Fuller mechanism, we exclude a fraction of 𝑓𝜈𝑠 ≲0.45, independent of mixing angle. Our results extend constraints on MWDM to a region of parameter space that has been relatively unconstrained with previous analysis.